Pinchoff by surface diffusion

Abstract

We construct surface diffusion flows $f:{\mathbb{T}}^2\times [0,T)\to {\mathbb{R}}^3$ that drive smooth closed embedded tori to pinchoff in finite time. The flow remains embedded for $t\in [0,T)$ and develops a curvature singularity only at a distinguished point $p$ as $t\nearrow T$. Away from $(p,T)$ the flow converges smoothly as $t\nearrow T$. We characterise the singularity profile: If $A(t)$ denotes the radius of the waist, and the surface is given locally near the waist by the rotation of a radial graph $(z,t)\mapsto r(z,t)$, then there exists a constant $\mu >0$ and smooth function $U:\mathbb{R}\to \mathbb{R}$ such that \[ A(t)=\{4μ(T-t)\}^{1/4}(1+o(1)), \qquad A(t)^{-1}r(A(t)ζ,t)\stackrel{C^\infty_{loc}}{\longrightarrow} U(ζ). \] Here $U$ is a rigorous realisation of the classical fundamental positive even conical similarity profile first computed numerically by Wong, Miksis, Voorhees and Davis and subsequently analysed by Bernoff, Bertozzi and Witelski.

arXiv comment48 pages, 1 figure